explicit answer error with dsolve

You are not doing anything wrong. Not all differential equations have analytic solutions.

Instead, create your differential equation as a system of first-order differential equations, and code it as an anonymous function to use in ode15s (which would be my choice for your problem). Two functions, odeToVectorField and matlabFunction can do this for you:

g = 9.81;                     % Gravitation (m/s^2)
C_D = 0.47;                   % Drag coefficient [-]
rho = 1.293;                  % Density of air [kg/m^3]
rho_steel = 8050;             % Density of steel [kg/m^3]
% %% system dimesions and properties 
r_ball=0.005;                               % Radius of the ball in [m] (ESTIMATED VALUE)
m_ball= rho_steel*(4/3 * pi * r_ball^3);    % Mass of the ball [kg] (ESTIMATED VALUE)
a = 0.005;                                   % a variable of the tube [m]
b = 0.008;                                  % b variable of the tube [m]
L = 0.112;                                  % length of the tube [m]  
h = 0.07                                    % equilibrium height
phi = 1.81*10^-5;                           % Viscosity of air [pa * s]
A_float = 0.0001327                         % Area of the float [m^2]
% %% Variables
Q=0.1;                       % Volume flow generated by compressor [m^3/s](ESTIMATED VALUE)

% Create differential equation
syms x(t) Y T 
dx = diff(x,t);
dxdx = diff(x,t,2);
ode = m_ball * diff(x,t,2) - 1/2 * rho * A_float * C_D * ((Q/((((((((b-a)*(h*x))/L)+a)^2)*pi)-A_float)))^2) 
+ g * m_ball == 0;

[VF,Subs] = odeToVectorField(ode)
odefcn = matlabFunction(VF, 'Vars',{T,Y})

producing:

VF =

                                                                                                           Y[2]
   46487591893913/(485952971979340400*(pi*((3*Y[1])/1600 + 1/200)^2 - 4895765877162515/36893488147419103232)^2)

Subs =

    x
   Dx

odefcn =

    function_handle with value:

      @(T,Y)[Y(2);1.0./(pi.*(Y(1).*1.875e-3+1.0./2.0e2).^2-1.327e-4).^2.*9.566273811345134e-5]

I always get the second output from odeToVectorField, because I then use it with legend in the plot of the ode15s result to know what the solutions represent.